The phenomenon of quantum superposition manifests in two distinct ways: it either spreads out across non-orthogonal basis states or remains concealed within their overlaps. Despite its profound implications, the resource theory of superposition often neglects the quantum superposition residing within these overlaps. However, this component is intricately linked to a form of state indistinguishability and can give rise to quantum correlations. In this paper, we introduce a pseudo-Hermitian representation of the density operator, wherein its diagonal elements correspond to biorthogonal extensions of Kirkwood-Dirac quasi-probabilities. This representation provides a unified framework for the inter-basis quantum superposition and basis state indistinguishability, giving rise to what we term as \textit{genuine} quantum superposition. Moreover, we propose appropriate generalizations of current superposition measures to quantify genuine quantum superposition that serves as the fundamental notion of nonclassicality from which both quantum coherence and correlations emerge. Finally, we explore potential applications of our theoretical framework, particularly in the quantification of electron delocalization in chemical bonding and aromaticity.
The phenomenon of quantum superposition manifests in two distinct ways: it either spreads out across non-orthogonal basis states or remains concealed within their overlaps. Despite its profound implications, the resource theory of superposition often neglects the quantum superposition residing within these overlaps. However, this component is intricately linked to a form of state indistinguishability and can give rise to quantum correlations. In this paper, we introduce a pseudo-Hermitian representation of the density operator, wherein its diagonal elements correspond to biorthogonal extensions of Kirkwood-Dirac quasi-probabilities. This representation provides a unified framework for the inter-basis quantum superposition and basis state indistinguishability, giving rise to what we term as \textit{genuine} quantum superposition. Moreover, we propose appropriate generalizations of current superposition measures to quantify genuine quantum superposition that serves as the fundamental notion of nonclassicality from which both quantum coherence and correlations emerge. Finally, we explore potential applications of our theoretical framework, particularly in the quantification of electron delocalization in chemical bonding and aromaticity.
This letter investigates the applications and extensions of the resource theory of quantum superposition within the realm of quantum chemistry. Specifically, we explore aromaticity, a fundamental concept originally developed to elucidate the structural symmetry, energetic stability, and chemical reactivity of benzene and its derivatives. While aromaticity and its counterpart, antiaromaticity, are associated with the delocalization of electrons between nonorthogonal atomic orbitals, they lack a universally accepted and comprehensive definition. We demonstrate that the genuine quantum superposition exhibited by biorthogonal atomic orbitals effectively captures the aromaticity order of representative monocyclic molecules. These findings reveal that the quantum resource theories hold significant implications, offering fresh insights into our comprehension of chemical bonding phenomena.
One of the fundamental features that separates quantum physics from classical physics is the idea of quantum superposition, also known as coherence. This thesis concentrates on understanding quantum coherence in the mathematical framework of resource theories, viewing it both as a resource to be harnessed and as a way to quantitatively characterise quantum states in contrast to classical states. We first examine the type of coherence resource theory which has emerged recently to cope with general settings where the physical nature of the medium encoding information is not crucial, such as computation. We identify the set of quantum processes in which coherence is neither created nor used, and use these to provide a physically motivated resource theory pictured in terms of interferometry. Using the same concepts, we then find connections between coherence and discord, a type of quantum correlation. In particular, we show how coherence can be used to generate discord, and explore basis-dependent discord as an intermediate quantity. The second part of the thesis applies the resource theory framework to quantify quantum macroscopicity, taken here to mean the extent to which coherence exists in a system on a macroscopic scale. We find the appropriate type of resource theory for this purpose, giving criteria for good measures of macroscopic coherence. We use these criteria to evaluate some previously proposed measures and highlight the role of the quantum Fisher information. Next, we build up measures based on the concept of macroscopic distinguishability and use them to show that macroscopic quantum states are fragile to noise induced by interaction with an environment. Finally, we apply measures based on the Fisher information to a range of experiments involving mechanical degrees of freedom, in order to compare their macroscopicity.
Abstract As an extension of classical probability distribution, the Kirkwood-Dirac distribution (KDD) was discussed by Kirkwood in 1933 and Dirac 1945, independently. Recently, it has been proved that nonclassical values (negative and non-real values) of the KDD have the ability of outperforming their classical counterparts in quantum computation, quantum measurement and so on. In this work, by dividing quantum states into KD-real (KD-free) and KD-imaginary (KD-resource) ones based on the KDD of a state, we establish a resource theory for KD-imaginarity with respect to a pair of bases (A, B), called the resource theory of Kirkwood-Dirac imaginarity. This theory is different from the resource theory of imaginarity of quantum states with respect to one basis A, where the free states are those that have real density matrices under the basis A.
The superposition principle lies at the heart of many nonclassical properties of quantum mechanics. Motivated by this, we introduce a rigorous resource theory framework for the quantification of superposition of a finite number of linear independent states. This theory is a generalization of resource theories of coherence. We determine the general structure of operations which do not create superposition, find a fundamental connection to unambiguous state discrimination, and propose several quantitative superposition measures. Using this theory, we show that trace decreasing operations can be completed for free which, when specialized to the theory of coherence, resolves an outstanding open question and is used to address the free probabilistic transformation between pure states. Finally, we prove that linearly independent superposition is a necessary and sufficient condition for the faithful creation of entanglement in discrete settings, establishing a strong structural connection between our theory of superposition and entanglement theory.
Abstract Just a few years after the inception of quantum mechanics, there has been a research program using the nonclassical values of some quasiprobability distributions to delineate the nonclassical aspects of quantum phenomena. In particular, in Kirkwood–Dirac (KD) quasiprobability distribution, the distinctive quantum mechanical feature of noncommutativity which underlies many nonclassical phenomena, manifests in the nonreal values and/or the negative values of the real part. Here, we develop a faithful quantifier of quantum coherence based on the KD nonclassicality which captures simultaneously the nonreality and the negativity of the KD quasiprobability. The KD-nonclassicality coherence thus defined, is upper bounded by the uncertainty of the outcomes of measurement described by a rank-1 orthogonal projection-valued measure (PVM) corresponding to the incoherent orthonormal basis which is quantified by the Tsallis 1 2 -entropy. Moreover, they are identical for pure states so that the KD-nonclassicality coherence for pure state admits a simple closed expression in terms of measurement probabilities. We then use the Maassen–Uffink uncertainty relation for min-entropy and max-entropy to obtain a lower bound for the KD-nonclassicality coherence of a pure state in terms of optimal guessing probability in measurement described by a PVM noncommuting with the incoherent orthonormal basis. We also derive a trade-off relation for the KD-nonclassicality coherences of a pure state relative to a pair of noncommuting orthonormal bases with a state-independent lower bound. Finally, we sketch a variational scheme for a direct estimation of the KD-nonclassicality coherence based on weak value measurement and thereby discuss its relation with quantum contextuality.
One of the fundamental features of quantum mechanics is the superposition principle, a manifestation of which is embodied in quantum coherence. Coherence of a quantum state is invariably defined with respect to a preferred set of pointer states, and there exist quantum coherence measures with respect to deterministically as well as probabilistically distinguishable sets of quantum state vectors. Here we study the resource theory of quantum coherence with respect to an arbitrary set of quantum state vectors, that may not even be probabilistically distinguishable. Geometrically, a probabilistically indistinguishable set of quantum state vectors forms a linearly dependent set. In quantum optics, the coherent states form an ``overcomplete basis'' of linearly dependent states and are useful in dealing with states that can be prepared in optical systems. Also, the resource theory of magic can be looked upon as a resource theory of quantum coherence with respect to a set of basis vectors that are probabilistically indistinguishable. These motivate us to consider a resource theory of coherence with respect to probabilistically indistinguishable pointers. We find the free states of the resource theory, and analyze the corresponding free operations, obtaining a necessary condition for an arbitrary quantum operation to be free. We identify a class of measures of the quantum coherence and, in particular, establish the monotonicity property of the measures. We find a connection of an arbitrary set of quantum state vectors with positive operator-valued measurements with respect to the resource theory being considered, which paves the way for an alternate definition of the free states. We subsequently examine the wave-particle duality in a double-slit setup in which superposition of probabilistically indistinguishable quantum state vectors is possible. Specifically, we report a complementary relation between quantum coherence and path distinguishability in such a setup.